1 60 J. W. Gibhs — Equilihrium of Heterogeneoiis Substances. 



In determining for any given positive values of T and P and any- 

 given values whatever of 3/,, M^, . . . M„ whether the expression 

 (133) is capable of a negative value for any phase of the components 

 aSj, aS'o, . . . /8„, and if not, whether it is capable of the value zero 

 for any other phase than that of which the stability is in question, it 

 is only necessary to consider phases having the temperature T and 

 pressure P. For we may assume that a mass of matter represented 

 by any values of m^, m^, • • • m„is capable of at least one state ot 

 not unstable equilibrium (which may or may not be a homogeneous 

 state) at this temperature and pressure. It may easily be shown 

 that for such a state the value of e — T?^-^ Pv must be as small as 

 for any other state of the same matter. The same will therefore be 

 true of the value of (133), Therefore if this expression is capable of 

 a negative value for any mass whatever, it will have a negative value 

 for that mass at the temperature T and pressure P. And if this 

 mass is not homogeneous, the value of (133) must be negative for at 

 least one of its homogeneous parts. So also, if the expression (133) is 

 not capable of a negative value for any phase of the comj)onents, 

 any phase for which it has the value zero must have the temperature 

 T and the pressure P. 



It may easily be shown that the same must be true in the limiting- 

 cases in which T=.0 and P=:0. For negative values of P, (133) 

 is always capable of negative values, as its value for a vacuum is Pv. 



For any body of the temperature T and pressure P, the expression 

 (133) may by (91) be reduced to the form 



t, — J/i m, — 31^ m^ ... —M„m„. (135) 



We have already seen (pages 131, 132) that an expression like 

 (133), when T, P, Jifj, J/g, . . . J/„ and v have any given finite 

 values, cannot have an infinite negative value as applied to any real 

 body. Hence, in determining whether (133) is capable of a negative 

 value for any phase of the components aS'j, S^, . . . jS„, and if not, 

 whether it is capable of the value zero for any other phase than that 

 of which the stability is in question, we have only to consider the 

 least value of which it is capable for a constant value of v. Any 

 body giving this value must satisfy the condition that for constant 

 volume 



de - T(h/ — J/, dm^ — J/^ dot^ ... — 3f„dm„^ 0, (136) 



stability of the fluid for constant temperature, or for constant pressure, or for both. 

 The number of coexistent phases will sometimes exceed by one or two the number of 

 the remaining equations, and then the equilibrium of the fluid will be neutral in 

 respect to one or two independent changes. 



